Round Tree Manor is a hotel that provides two types of rooms with three rental classes: Super Saver, Deluxe, and Business. The profit per night for each type of room and rental class is as follows:

Rental Class

Super Saver Deluxe Business

Room Type I $($Mountain View$)$ $$30$ $$35-$

Type II $($Street View$)$ $$15$ $$25$ $$35$

Round Tree's management makes a forecast of the demand by rental class for each night in the future. A linear programming model developed to maximize profit is used to determine how many reservations to accept for each rental class. The demand forecast for a particular night is $130$ rentals in the Super Saver class, $65$ in the Deluxe class, and $45$ in the Business class. Since these are the forecasted demands, Round Tree will take no more than these amounts of each reservation for each rental class. Round Tree has a limited number of each type of room. There are $90$ Type I rooms and $130$ Type II rooms.

$($a$)$ Formulate and solve a linear program to determine how many reservations to accept in each rental class and how the reservations should be allocated to room types. If an amount is zero, enter $0.$

Rental Class with room type No$.$ of Reservations

Super Saver rentals allocated to room type I

Super Saver rentals allocated to room type II

Deluxe rentals allocated to room type I

Deluxe rentals allocated to room type II

Business rentals allocated to room type II

$($b$)$ For the solution in part $($a$),$ how many reservations can be accommodated in each rental class?

Rental Class No$.$ of Reservations

Super Saver

Deluxe

Business

Demand for $-$ Select your answer $-$DeluxeSuper SaverBusinessItem $9$ rental class was not satisfied.

$($c$)$ With a little work, an unused office area could be converted to a rental room. If the conversion cost is the same for both types of rooms, would you recommend converting the office to a Type I or a Type II room?

Type I Type II

Shadow Price $ $

Convert an unused office area to $-$ Select your answer $-$Type IType IIItem $12$ room.

Explain.

Converting the unused office area to this type of room increases profit by $ $.$

$($d$)$ Could the linear programming model be modified to plan for the allocation of rental demand for the next night?

$-$ Select your answer $-$YesNoItem $14$

What information would be needed and how would the model change? Explain.

$($i$)$ We would need a forecast of demand for each rental class on the next night to use as the right$-$hand sides of the first three constraints

$($ii$)$ We would need to know if Type $1$ rooms can be used as Business class rooms the next night and add a variable to the objective function.

$($iii$)$ We would need to know how many rooms of Type I and Type II there will be on the next night to use as the right$-$hand sides of the last two constraints.

$($iv$)$ We would need to know whether the profit per night of each type of room and rental class will change and use these as objective coefficients.

$-$ Select your answer $-$Option $($i$)$Option $($ii$)$Option $($iii$)$Option $($iv$)$Item $15$